Both forms of negation are correct. The former is known as no-negation:

Dr Gillian McKeith has no medical qualification

whereas the latter is known as not-negation:

Dr Gillian McKeith doesn't have a medical qualification

The "Longman Student Grammar of Spoken and Written English" contains a brief section (8.8.8) that discusses choosing between not-negation and no-negation. It affirms that not-negation is much more common, and that no-negation is rare in conversation and moderately more common in writing.

The choice between not-negation and no-negation may also carry in some cases an additional meaning. For example, when I write:

She isn't a medical doctor

I'm simply stating that she doesn't hold a medical degree. But when I write:

She is no medical doctor

what I'm actually saying is that although one could have thought that Dr Gillian held a medical degree, she actually doesn't.

Both forms of negation are correct. The former is known as no-negation:

Dr Gillian McKeith has no medical qualification

whereas the latter is known as not-negation:

Dr Gillian McKeith doesn't have a medical qualification

The "Longman Student Grammar of Spoken and Written English" contains a brief section (8.8.8) that discusses choosing between not-negation and no-negation. It affirms that not-negation is much more common, and that no-negation is rare in conversation and moderately more common in writing.

The choice between not-negation and no-negation may also carry in some cases additional meaning. For example, when I write:

She isn't a medical doctor

I'm simply stating that she doesn't hold a medical degree. But when I write:

She is no medical doctor

what I'm actually saying is that although one could have thought that Dr Gillian held a medical degree, she actually doesn't.